External ray
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An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set.[1] Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a ray.
External rays are used in complex analysis, particularly in complex dynamics and geometric function theory.
History
[edit]External rays were introduced in Douady and Hubbard's study of the Mandelbrot set
Types
[edit]Criteria for classification :
plane
[edit]External rays of (connected) Julia sets on dynamical plane are often called dynamic rays.
External rays of the Mandelbrot set (and similar one-dimensional connectedness loci) on parameter plane are called parameter rays.
bifurcation
[edit]Dynamic ray can be:
When the filled Julia set is connected, there are no branching external rays. When the Julia set is not connected then some external rays branch.[5]
stretching
[edit]Stretching rays were introduced by Branner and Hubbard:[6][7]
"The notion of stretching rays is a generalization of that of external rays for the Mandelbrot set to higher degree polynomials."[8]
landing
[edit]Every rational parameter ray of the Mandelbrot set lands at a single parameter.[9][10]
Maps
[edit]Polynomials
[edit]Dynamical plane = z-plane
[edit]External rays are associated to a compact, full, connected subset of the complex plane as :
- the images of radial rays under the Riemann map of the complement of
- the gradient lines of the Green's function of
- field lines of Douady-Hubbard potential[11]
- an integral curve of the gradient vector field of the Green's function on neighborhood of infinity[12]
External rays together with equipotential lines of Douady-Hubbard potential ( level sets) form a new polar coordinate system for exterior ( complement ) of .
In other words the external rays define vertical foliation which is orthogonal to horizontal foliation defined by the level sets of potential.[13]
Uniformization
[edit]Let be the conformal isomorphism from the complement (exterior) of the closed unit disk to the complement of the filled Julia set .
where denotes the extended complex plane. Let denote the Boettcher map.[14] is a uniformizing map of the basin of attraction of infinity, because it conjugates on the complement of the filled Julia set to on the complement of the unit disk:
and
A value is called the Boettcher coordinate for a point .
Formal definition of dynamic ray
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/79/Erays.svg/220px-Erays.svg.png)
The external ray of angle noted as is:
- the image under of straight lines
- set of points of exterior of filled-in Julia set with the same external angle
Properties
[edit]The external ray for a periodic angle satisfies:
and its landing point[15] satisfies:
Parameter plane = c-plane
[edit]"Parameter rays are simply the curves that run perpendicular to the equipotential curves of the M-set."[16]
Uniformization
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/35/Jung200.png/220px-Jung200.png)
![](https://upload.wikimedia.org/wikipedia/commons/thumb/6/60/Jung50e.png/220px-Jung50e.png)
Let be the mapping from the complement (exterior) of the closed unit disk to the complement of the Mandelbrot set .[17]
and Boettcher map (function) , which is uniformizing map[18] of complement of Mandelbrot set, because it conjugates complement of the Mandelbrot set and the complement (exterior) of the closed unit disk
it can be normalized so that :
where :
- denotes the extended complex plane
Jungreis function is the inverse of uniformizing map :
In the case of complex quadratic polynomial one can compute this map using Laurent series about infinity[20][21]
where
Formal definition of parameter ray
[edit]The external ray of angle is:
- the image under of straight lines
- set of points of exterior of Mandelbrot set with the same external angle [22]
Definition of the Boettcher map
[edit]Douady and Hubbard define:
so external angle of point of parameter plane is equal to external angle of point of dynamical plane
External angle
[edit]-
collecting bits outwards
-
Binary decomposition of unrolled circle plane
-
binary decomposition of dynamic plane for f(z) = z^2
Angle
Principal value of external angles are measured in turns modulo 1
Compare different types of angles :
- external ( point of set's exterior )
- internal ( point of component's interior )
- plain ( argument of complex number )
external angle | internal angle | plain angle | |
---|---|---|---|
parameter plane | |||
dynamic plane |
Computation of external argument
[edit]- argument of Böttcher coordinate as an external argument[24]
- kneading sequence as a binary expansion of external argument[25][26][27]
Transcendental maps
[edit]For transcendental maps ( for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism.[28][29]
Here dynamic ray is defined as a curve :
- connecting a point in an escaping set and infinity [clarification needed]
- lying in an escaping set
Images
[edit]Dynamic rays
[edit]- unbranched
-
Julia set for with 2 external ray landing on repelling fixed point alpha
-
Julia set and 3 external rays landing on fixed point
-
Dynamic external rays landing on repelling period 3 cycle and 3 internal rays landing on fixed point
-
Julia set with external rays landing on period 3 orbit
-
Rays landing on parabolic fixed point for periods 2-40
- branched
-
Branched dynamic ray
Parameter rays
[edit]Mandelbrot set for complex quadratic polynomial with parameter rays of root points
-
External rays for angles of the form : n / ( 21 - 1) (0/1; 1/1) landing on the point c= 1/4, which is cusp of main cardioid ( period 1 component)
-
External rays for angles of the form : n / ( 22 - 1) (1/3, 2/3) landing on the point c= - 3/4, which is root point of period 2 component
-
External rays for angles of the form : n / ( 23 - 1) (1/7,2/7) (3/7,4/7) landing on the point c= -1.75 = -7/4 (5/7,6/7) landing on the root points of period 3 components.
-
External rays for angles of form : n / ( 24 - 1) (1/15,2/15) (3/15, 4/15) (6/15, 9/15) landing on the root point c= -5/4 (7/15, 8/15) (11/15,12/15) (13/15, 14/15) landing on the root points of period 4 components.
-
External rays for angles of form : n / ( 25 - 1) landing on the root points of period 5 components
-
internal ray of main cardioid of angle 1/3: starts from center of main cardioid c=0, ends in the root point of period 3 component, which is the landing point of parameter (external) rays of angles 1/7 and 2/7
-
Internal ray for angle 1/3 of main cardioid made by conformal map from unit circle
-
Mini Mandelbrot set with period 134 and 2 external rays
-
Wakes near the period 3 island
-
Wakes along the main antenna
Parameter space of the complex exponential family f(z)=exp(z)+c. Eight parameter rays landing at this parameter are drawn in black.
Programs that can draw external rays
[edit]- Mandel - program by Wolf Jung written in C++ using Qt with source code available under the GNU General Public License
- Java applets by Evgeny Demidov ( code of mndlbrot::turn function by Wolf Jung has been ported to Java ) with free source code
- ezfract by Michael Sargent, uses the code by Wolf Jung
- OTIS by Tomoki KAWAHIRA - Java applet without source code
- Spider XView program by Yuval Fisher
- YABMP by Prof. Eugene Zaustinsky Archived 2006-06-15 at the Wayback Machine for DOS without source code
- DH_Drawer Archived 2008-10-21 at the Wayback Machine by Arnaud Chéritat written for Windows 95 without source code
- Linas Vepstas C programs for Linux console with source code
- Program Julia by Curtis T. McMullen written in C and Linux commands for C shell console with source code
- mjwinq program by Matjaz Erat written in delphi/windows without source code ( For the external rays it uses the methods from quad.c in julia.tar by Curtis T McMullen)
- RatioField by Gert Buschmann, for windows with Pascal source code for Dev-Pascal 1.9.2 (with Free Pascal compiler )
- Mandelbrot program by Milan Va, written in Delphi with source code
- Power MANDELZOOM by Robert Munafo
- ruff by Claude Heiland-Allen
See also
[edit]![](https://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
- external rays of Misiurewicz point
- Orbit portrait
- Periodic points of complex quadratic mappings
- Prouhet-Thue-Morse constant
- Carathéodory's theorem
- Field lines of Julia sets
References
[edit]- ^ J. Kiwi : Rational rays and critical portraits of complex polynomials. Ph. D. Thesis SUNY at Stony Brook (1997); IMS Preprint #1997/15. Archived 2004-11-05 at the Wayback Machine
- ^ Inou, Hiroyuki; Mukherjee, Sabyasachi (2016). "Non-landing parameter rays of the multicorns". Inventiones Mathematicae. 204 (3): 869–893. arXiv:1406.3428. Bibcode:2016InMat.204..869I. doi:10.1007/s00222-015-0627-3. S2CID 253746781.
- ^ Atela, Pau (1992). "Bifurcations of dynamic rays in complex polynomials of degree two". Ergodic Theory and Dynamical Systems. 12 (3): 401–423. doi:10.1017/S0143385700006854. S2CID 123478692.
- ^ Petersen, Carsten L.; Zakeri, Saeed (2020). "Periodic Points and Smooth Rays". arXiv:2009.02788 [math.DS].
- ^ Holomorphic Dynamics: On Accumulation of Stretching Rays by Pia B.N. Willumsen, see page 12
- ^ The iteration of cubic polynomials Part I : The global topology of parameter by BODIL BRANNER and JOHN H. HUBBARD
- ^ Stretching rays for cubic polynomials by Pascale Roesch
- ^ Komori, Yohei; Nakane, Shizuo (2004). "Landing property of stretching rays for real cubic polynomials" (PDF). Conformal Geometry and Dynamics. 8 (4): 87–114. Bibcode:2004CGDAM...8...87K. doi:10.1090/s1088-4173-04-00102-x.
- ^ A. Douady, J. Hubbard: Etude dynamique des polynˆomes complexes. Publications math´ematiques d’Orsay 84-02 (1984) (premi`ere partie) and 85-04 (1985) (deuxi`eme partie).
- ^ Schleicher, Dierk (1997). "Rational parameter rays of the Mandelbrot set". arXiv:math/9711213.
- ^ Video : The beauty and complexity of the Mandelbrot set by John Hubbard ( see part 3 )
- ^ Yunping Jing : Local connectivity of the Mandelbrot set at certain infinitely renormalizable points Complex Dynamics and Related Topics, New Studies in Advanced Mathematics, 2004, The International Press, 236-264
- ^ POLYNOMIAL BASINS OF INFINITY LAURA DEMARCO AND KEVIN M. PILGRIM
- ^ How to draw external rays by Wolf Jung
- ^ Tessellation and Lyubich-Minsky laminations associated with quadratic maps I: Pinching semiconjugacies Tomoki Kawahira Archived 2016-03-03 at the Wayback Machine
- ^ Douady Hubbard Parameter Rays by Linas Vepstas
- ^ John H. Ewing, Glenn Schober, The area of the Mandelbrot Set
- ^ Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. Volume 52, Number 4 (1985), 935-938.
- ^ Adrien Douady, John Hubbard, Etudes dynamique des polynomes complexes I & II, Publ. Math. Orsay. (1984-85) (The Orsay notes)
- ^ Bielefeld, B.; Fisher, Y.; Vonhaeseler, F. (1993). "Computing the Laurent Series of the Map
Ψ : C − D → C − M". Advances in Applied Mathematics. 14: 25–38. doi:10.1006/aama.1993.1002. - ^ Weisstein, Eric W. "Mandelbrot Set." From MathWorld--A Wolfram Web Resource
- ^ An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira
- ^ http://www.mrob.com/pub/muency/externalangle.html External angle at Mu-ENCY (the Encyclopedia of the Mandelbrot Set) by Robert Munafo
- ^ Computation of the external argument by Wolf Jung
- ^ A. DOUADY, Algorithms for computing angles in the Mandelbrot set (Chaotic Dynamics and Fractals, ed. Barnsley and Demko, Acad. Press, 1986, pp. 155-168).
- ^ Adrien Douady, John H. Hubbard: Exploring the Mandelbrot set. The Orsay Notes. page 58
- ^ Exploding the Dark Heart of Chaos by Chris King from Mathematics Department of University of Auckland
- ^ Topological Dynamics of Entire Functions by Helena Mihaljevic-Brandt
- ^ Dynamic rays of entire functions and their landing behaviour by Helena Mihaljevic-Brandt
- Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer 1993
- Adrien Douady and John H. Hubbard, Etude dynamique des polynômes complexes, Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985)
- John W. Milnor, Periodic Orbits, External Rays and the Mandelbrot Set: An Expository Account; Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque No. 261 (2000), 277–333. (First appeared as a Stony Brook IMS Preprint in 1999, available as arXiV:math.DS/9905169.)
- John Milnor, Dynamics in One Complex Variable, Third Edition, Princeton University Press, 2006, ISBN 0-691-12488-4
- Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002
External links
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png)
- Hubbard Douady Potential, Field Lines by Inigo Quilez [permanent dead link]
- Intertwined Internal Rays in Julia Sets of Rational Maps by Robert L. Devaney
- Extending External Rays Throughout the Julia Sets of Rational Maps by Robert L. Devaney With Figen Cilingir and Elizabeth D. Russell
- John Hubbard's presentation, The Beauty and Complexity of the Mandelbrot Set, part 3.1 Archived 2008-02-26 at the Wayback Machine
- videos by ImpoliteFruit
- Milan Va. "Mandelbrot set drawing". Retrieved 2009-06-15.[permanent dead link]